Lay them on the right side or they will barf and maybe die on you if they are really out of it.
Older methods recommended laying on the left side because cause the stomach is on the left side - if the patient is on the right side the chances of them aspirating any of their vomit are higher. But it causes no harm if you put them on the right side as long as you monitor the airway closely. If a casualty has a wound or some cause for pain that is affected by which side the weight is on, one should prioritise that consideration. Monitoring breathing in an unconscious person should be normal anyway.
The esophagus joins the stomach so that if lying on the left side, most of the stomach is below the esophagus. If lying on the right side, the stomach is higher. None of that is to say that you will not puke if lying on the left. Puking is a complicated interplay of many body systems, the last of which is the respiratory muscles forcing the upper digestive tract into reverse. If lying on the left, the stomach contents simply go out the top instead of more easily out the bottom. It may make some small difference how much gets puked-up but it will not keep the puke response from happening.
Lying on the left does still have certain digestive benefits. Some people with gastric disorders find relief from left-side sleeping since the stomach's enzymes are better held low by gravity.
Gee, Sharonna, are we running out of questions today?
This is a CPR question. Life or death, young man.
Thank God, I do not feel like thinking.
Sounds good to me DTT
Yes. That. 
Whew!
No one was there for Jimi Hendrix in his time of need.
